When Foo Fighters' Dave Grohl and Taylor Hawkins inducted the veteran rock trio Rush into the Rock and Roll Hall of Fame this month, they did so with great affection and a winning mischievousness. For starters, they donned lengthy wigs and white silk kimonos, dressing as Rush circa 1976 to perform "Overture", the wonderfully grandiose opening track from the group's dystopian concept album, 2112.

That Grohl – modern rock's great tastemaker and Zelig figure – so relished inducting Rush is another mark of how far they have come. Formed in Toronto in 1968, Geddy Lee (bass and vocals), Alex Lifeson (guitar) and Neil Peart (drums) have sold more than 40 million albums, morphing from cherished outsiders to the biggest "cult" band in the world.

"For years, I think it was almost taboo for people to acknowledge the influence of such a freak-flag band as Rush," says Lee today ahead of the band's UK arena tour in May. "More recently, there's been a maturing of our audience into positions of success and power."

The band's mainstream acceptance – if that is what they are finally winning – has been a long time coming. Rolling Stone was sniffy about Rush for decades – and this despite a revelation from an insider there that the group were one of the magazine's most-requested cover stories. Lee's unusually high voice has also made things tricky at times, one critic famously describing him as sounding like "a chipmunk on acid".

"'The damned howling in Hades' was another one," the singer laughs, "but it suited what we were doing back then and I was never going to change that. The gibes were painful at first; I'm a human being. I just grew a thick skin and learnt to accept that my voice is part of what makes Rush unique."

In truth, what's really odd about Rush's story is the persistence of certain key myths. It's odd, for example, that a band that has so consistently sent itself up (witness Lifeson's surreal acceptance speech at the Rock and Roll Hall of Fame via YouTube, or try Googling Lee as the kilt-wearing Scotsman Harry Satchel) is still sometimes seen as a po-faced act for whom Samuel Taylor Coleridge remains a lyrical touchstone.

"Yeah, that prog-rock style of music we were doing early on was a bit overblown and if you write an 11-minute song called 'Xanadu' it's probably inevitable that the critics will say you're pretentious", Lee says.

"As for the lyrics, from [1980 album] Permanent Waves on, Neil was often writing very insightful stuff about the human condition. The early songs about Greek gods and mythical creatures have always got the most flash, but then there are later songs like 'Losing It' [about ageing's degenerative effect on artists' gifts], and 'Red Sector A' [part inspired by Lee's parents being survivors of Auschwitz]."

Rush's 19th studio album, Clockwork Angels, reached No 2 in the US last year; No 1 in the group's native Canada. It would also have reached No 1 in the UK had copies that came free with a Classic Rock magazine fan-pack edition fulfilled chart-eligibility criteria.

How, I ask Lee in closing, have Rush managed to avoid the sadder rock-star clichés and stay successful for so long?

"We've been able to sin in silence, and fortunately whatever transgressions there have been have slipped by relatively unnoticed," he says. "We all have healthy relationships, and there's something in our nature that's not for quitting, whether it's a band or a marriage. Maybe it's a Canadian thing."

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